Variational Method Research Articles

Parameter estimation for allometric trophic network models: A variational Bayesian inverse problem approach

Abstract Differential equation models are powerful tools for predicting biological systems, capable of projecting far into the future and incorporating data recorded at arbitrary times. However, estimating these models' parameters from observations can be challenging because numerical methods are often required to approximate their solution. An example of such a model is the allometric trophic network model, for which studies considering its inverse problem are limited, particularly in the Bayesian framework. Here we develop a variational Bayesian method for parameter inference of the allometric trophic network model and explore how accurately we can recover its parameter values. We represent the model as a Bayesian neural network, which combines an artificial neural network with Bayesian inference, using a surrogate for the posterior distribution of model parameters, and train this model by evolutionary optimization to avoid potentially costly computation of the gradient with respect to the model parameters. Using synthetic data, we compare the accuracy of this variational inference to ordinary least squares estimation. To reduce the number of estimated parameters, we focus on the inference of functional response parameters. Our variational Bayesian method yields parameter estimates that are comparable to the ordinary least squares results in terms of accuracy. The method provides a promising approach for including uncertainty quantification in parameter estimation, which the simple ordinary least squares approach as it is does not address. Regardless of the method, potential multimodality of the inference problem is nonetheless important to keep in mind. The present study suggests a technique for parameter inference of ordinary differential equation models in the Bayesian context. We propose the method especially for validation of the allometric trophic network model against empirical data.

Analytical solutions of the Caudrey-Dodd-Gibbon equation using Khater II and variational iteration methods.

This study focuses on solving the Caudrey-Dodd-Gibbon ( ) equation using the Khater II ( ) method and the Variational Iteration ( ) method. The equation is a pivotal mathematical model in nonlinear wave dynamics, essential for understanding the evolution, interaction, and preservation of wave forms in dispersive media. Its applications span various fields, including fluid dynamics, nonlinear optics, and plasma physics, where it plays a crucial role in analyzing solitons and complex wave interactions. In this research, we meticulously implement the and methods to derive solutions for this nonlinear partial differential equation. Our findings reveal new aspects of the equation's behavior, offering deeper insights into nonlinear wave phenomena. The significance of this study lies in its contribution to advancing the understanding of these phenomena and their practical applications in the academic realm. The results underscore the effectiveness of the employed methods, their innovative contributions, and their relevance to applied mathematics.

Dynamic plastic damage analysis of bending-torsion coupling in cross stiffener systems under mass impact

ABSTRACT This study proposes a simpler theoretical model to predict the bending-torsion coupling dynamic plastic response of cross stiffener systems under mass impact. Traditional methods often rely on torque balance and momentum conservation equations, which involve complex and difficult calculations. To simplify this process, the study uses the law of energy conservation and applies the variational method to solve the dynamic plastic damage of the cross stiffener system. The plastic response of a single stiffener is first analyzed and compared with existing theories, showing consistent results for maximum displacement. The study then extends the model to more complex conditions, performing comparative analyses on stiffeners of different sizes through numerical simulation. The findings show that the maximum displacement at the load and node remains within a reasonable range, with a displacement difference of only 0.9 mm and an error margin of 4.79% in one of the test conditions.

A robust and efficient cubature Kalman filter based on the variational Bayesian method and its application in target tracking

Abstract In this paper, we propose a robust cubature Kalman filter (CKF) for nonlinear state-space models with unknown state and measurement noise covariance matrix. We study situations in which sensors are independent of each other. Therefore, the unknown measurement noise variance is modeled as an unknown inverse gamma (IG) distribution. The Gaussian-Student-t-inverse-Wishart (GSTIW) mixture distribution is used to model the one-step prediction distribution. Modeling generates numerous unknown parameters. Therefore, we adopt the statistical linearization method to linearize the observation function and then estimate the state and parameters separately to reduce the computational burden of estimating unknown parameters, significantly improving the algorithm’s efficiency. Finally, using the variational Bayesian method, a novel and efficient robust CKF based on the IG distribution and GSTIW mixture distributions (IG-GSTIW-CKF) is obtained. Simulation and experimental results show that the proposed method has better estimation accuracy than several advanced algorithms when sensors are independent. In addition, the efficiency of the proposed algorithm is significantly higher than that of other CKF-based methods.

Data-Driven Morozov Regularization of Inverse Problems

The solution of inverse problems is crucial in various fields such as medicine, biology, and engineering, where one seeks to find a solution from noisy observations. These problems often exhibit non-uniqueness and ill-posedness, resulting in instability under noise with standard methods. To address this, regularization techniques have been developed to balance data fitting and prior information. Recently, data-driven variational regularization methods have emerged, mainly analyzed within the framework of Tikhonov regularization, termed Network Tikhonov (NETT). This paper introduces Morozov regularization combined with a learned regularizer, termed DD-Morozov regularization. Our approach employs neural networks to define non-convex regularizers tailored to training data, enabling a convergence analysis in the non-convex context with noise-dependent regularizers. We also propose a refined training strategy that improves adaptation to ill-posed problems compared to NETT’s original strategy, which primarily focuses on addressing non-uniqueness. We present numerical results for attenuation correction in photoacoustic tomography, comparing DD-Morozov regularization with NETT using the same trained regularizer, both with and without an additional total variation regularizer.

A Novel Approach for Time-Local Fractional Solutions of Certain Nonlinear Partial Differential Equations in Fractal Dimension

Time-local fractional approaches for nonlinear partial differential equations in fractal dimensions are essential for capturing the complex, irregular behaviors found in fractal systems. In this paper, a new modification of the local fractional Laplace variational iteration method (MLFLVIM) for obtaining analytical approximate solutions to the fractional gas dynamics equation, fractional Stefan equation, and fractional Newell-Whitehead-Segel equation within the context of fractal time space is presented. The proposed method (MLFLVIM) elegantly combines the local fractional Laplace transform (LFLT) with modified variational iteration method. Specifically, we first apply the (LFLT) to the given local fractional PDEs, yielding a transformed system of equations. We then apply modified variational iteration to this system. Finally, we use the inverse of (LFLT) to obtain the desired solution. To demonstrate the effectiveness of this approach, we implement it on three numerical physical problems. The results show that the (MLFLVIM) can successfully handle these nonlinear LFPDEs and provide accurate analytical approximation solutions.

Multi‐bump solutions for the nonlinear magnetic Schrödinger equation with logarithmic nonlinearity

AbstractIn this paper, we study the following nonlinear magnetic Schrödinger equation with logarithmic nonlinearity where the magnetic potential , is a parameter and the nonnegative continuous function has the deepening potential well. Using the variational methods, we obtain that the equation has at least multi‐bump solutions when is large enough.

Energy absorption characteristics and auxetic effect of novel elliptic-arc re-entrant honeycomb structures

The novel elliptical-arc re-entrant honeycomb (ERH) is designed by substituting the straight inclined struts within the re-entrant honeycomb with elliptical-arc struts. To assess its performance effectively, the study established the 3D equivalent Cauchy model (3D-ECM) and 2D equivalent Kirchhoff–Love model (2D-EKM) using the variational asymptotic method. The equivalent properties derived from the unit-cell constitutive model were integrated into the equivalent models for macroscopic analysis. Through 3D printing experiments and numerical simulations, the model’s accuracy in predicting the compression behaviors and auxetic effects of various uni- and multi-cellular ERHs under uniaxial compression, as well as the three-point bending behaviors of ERH panels were confirmed. In addition, this model substantially simplifies the modeling process, leading to a 8-fold increase in computational efficiency. Parametric analyses demonstrated that the ERH structure can uphold a beneficial auxetic effect while achieving lightweight and high strength characteristics when the axial ratio of the ellipse equals 1.25. Furthermore, ERH structures outperform arc-shaped re-entrant and regular re-entrant honeycombs in energy absorption and specific energy absorption capacity. These findings offer valuable insights for the preliminary design and optimization of ERH structures.

Optimizing quantitative photoacoustic imaging systems: The Bayesian Cramér-Rao bound approach

Abstract Quantitative photoacoustic computed tomography (qPACT) is an emerging medical imaging modality that carries the promise of high-contrast, fine-resolution imaging of clinically relevant quantities like hemoglobin concentration and blood-oxygen saturation. However, qPACT image reconstruction is governed by a multiphysics, partial differential equation (PDE) based inverse problem that is highly non-linear and severely ill-posed. Compounding the di culty of the problem is the lack of established design standards for qPACT imaging systems, as there is currently a proliferation of qPACT system designs for various applications and it is unknown which ones are optimal or how to best modify the systems under various design constraints. This work introduces a novel computational approach for the optimal experimental design (OED) of qPACT imaging systems based on the Bayesian Cramér-Rao bound (CRB). Our approach incorporates several techniques to address challenges associated with forming the bound in the infinite-dimensional function space setting of qPACT, including priors with trace-class covariance operators and the use of the variational adjoint method to compute derivatives of the log-likelihood function needed in the bound computation. The resulting Bayesian CRB based design metric is computationally efficient and independent of the choice of estimator used to solve the inverse problem. The efficacy of the bound in guiding experimental design was demonstrated in a numerical study of qPACT design schemes under a stylized two-dimensional imaging geometry. To the best of our knowledge, this is the first work to propose a Bayesian CRB-based design for systems governed by PDEs.


Semantic information guided diffusion posterior sampling for remote sensing image fusion

The task of image fusion for optical images and SAR images is to integrate valuable information from source images. Recently, owing to powerful generation, diffusion models, e.g., diffusion denoising probabilistic model and score-based diffusion model, are flourished in image processing, and there are some effective attempts in image fusion by scholars’ progressive explorations. However, the diffusion models for image fusion suffer from inevitable SAR speckle that seriously shelters from effective information in the same location of optical image. Besides, these methods integrate pixel-level features without information for high-level tasks, e.g., target detection and image classification, which leads fused images are insufficient and their application accuracies are low, for high-level tasks. To tackle these hurdles, we propose the semantic information guided diffusion posterior sampling for image fusion. Firstly, we employ the SAR-BM3D as preprocessing to despeckle. Then, the sampling model is established with fidelity, regularization and semantic information guidance term. The first two terms are obtained by the variational diffusion method via variational inference and first-order stochastic optimization. The last term is served by cross entropy loss between annotation and classification result from FLCNet we design. Finally, the experiments validate the feasibility and superiority of the proposed method on WHU-OPT-SAR dataset and DDHRNet dataset.

QUALITATIVE ANALYSIS OF TEMPERATURE ERROR OF NEGATIVE PRESSURE WAVE METHOD FOR DETERMINING THE LOCATION OF A PIPELINE LEAK

The article is devoted to the qualitative analysis of the temperature error of the negative pressure wave method for determining the location of a pipeline leak. Recording the arrival time of the negative wave at both ends of the pipeline allows you to determine the location of a pipeline leak. In contrast to known works on this topic, this article proposes a method for qualitative analysis of the error in determining a leak location. Based on the variational optimization method, an optimal ratio of the mode indicators at which the minimum error in determining the location of a leak can be achieved is defined. A method for qualitative analysis of the error in determining the leak location using the negative pressure difference wave method is proposed. The optimal correlation of the mode functions is determined at which the minimum error in determining the leak location is achieved. An expression for estimating the relative error in determining the leak location using the negative pressure difference wave method is obtained.

Variational Hirshfeld Partitioning: General Framework and the Additive Variational Hirshfeld Partitioning Method.

We introduce the general mathematical framework of variational Hirshfeld partitioning, wherein the best possible approximation to a molecule's electron density is obtained by minimizing the f-divergence between the molecular density and a non-negative linear combination of (normalized) basis functions. This framework subsumes several existing methods that variationally optimize their pro-atoms, like (Gaussian) iterative stockholder analysis (ISA and GISA) and minimal basis iterative stockholder partitioning (MBIS), and provides a solid foundation for developing mathematically rigorous partitioning schemes. In this paper, we delve into the mathematical underpinnings of Hirshfeld-inspired partitioning schemes and show that among all the valid f-divergence measures only the extended Kullback-Leibler is a suitable choice. This led us to develop a novel partitioning scheme, called additive variational Hirshfeld (AVH), which constructs the pro-molecular density as a convex linear combination of the densities from selected states of isolated atoms and atomic ions. The AVH method is size-consistent with a unique solution and provides a straightforward approach for adding constraints for fragment properties. It also results in an intuitively appealing valence-bond-like decomposition of the molecular density as a weighted average of the densities of the atomic states in the molecule; that is, the AVH atomic density is a minimal deformation of the corresponding isolated atomic reference state's density. Compared to other variational Hirshfeld variants, our numerical results show that AVH yields chemically interpretable and sensible atomic charges that are not excessively large and demonstrate computational robustness.

Nuclear norm regularized loop optimization for tensor network

We propose a loop optimization algorithm based on nuclear norm regularization for the tensor network. The key ingredient of this scheme is to introduce a rank penalty term proposed in the context of data processing. Compared to the standard variational periodic matrix product states method, this algorithm can circumvent the local minima related to short-ranged correlation in a simpler fashion. We demonstrate its performance when used as a part of the tensor network renormalization algorithms [S. Yang, Z.-C. Gu, and X.-G. Wen, ] for the critical two-dimensional Ising model. The scale invariance of the renormalized tensors is attained with higher accuracy while the higher parts of the scaling dimension spectrum are obtained in a more stable fashion. Published by the American Physical Society 2024

Two Positive Solutions for Elliptic Differential Inclusions

The existence of two positive solutions for an elliptic differential inclusion is established, assuming that the nonlinear term is an upper semicontinuous set-valued mapping with compact convex values having subcritical growth. Our approach is based on variational methods for locally Lipschitz functionals. As a consequence, a multiplicity result for elliptic Dirichlet problems having discontinuous nonlinearities is pointed out.

Mathematical Models for Removal of Pharmaceutical Pollutants in Rehabilitated Treatment Plants

This paper aims to investigate appropriate mathematical models devoted to the optimization of some cleaning processes related to pharmaceutical contaminant removal. In our recent works, we found the rehabilitation of the existing cleaning plants as a viable solution for the removal of this type of micropollutants from waters by introducing efficient techniques such as adsorption on granulated active carbon filters and micro-, nano-, or ultrafiltration. To have these processes under better control and to assure the transfer from small- to large-scale treatment stations, specific mathematical models are necessary. Starting from Navier–Stokes equations and imposing proper boundary conditions, some mathematical physics problems are obtained for which adequate solving methods via variational methods and surjectivity results are proposed. The importance of these solution characterizations consists in their continuation in adequate numerical methods and the possibility to visualize the result by using a CFD program.

Morse index for solutions of a nonlinear Kirchhoff equation

In this paper, we study a perturbed nonlinear Kirchhoff equation with subcritical growth in R3. Although the existence of concentrated solutions with a single peak or multi peaks to the problem above has been obtained in Li et al. [J. Differ. Equations 268, 541–589 (2020)] and Luo et al. [Proc. R. Soc. Edinburgh, Sect. A 149, 1097–1122 (2019)], respectively, the Morse indices of them remain open. First, we compute the Morse index of single-peak solutions concentrated at a point P∈R3 by variational methods, which can also be applied to the case where P is a degenerate critical point of V. Then, we also study the Morse index of multi-peak solutions concentrated at the non-degenerate critical points of V. Here the main difficulty comes from the nonlocal term ∫R3|∇u|2dyΔu. In addition, since the estimates of the eigenvalues and the eigenfunctions of the linearized problem associated to the limit problem are unknown, we have to study them independently, which are quite interesting.

An Efficient Method for Batch Derivation of Detached Eclipsing Binary Parameters: Analysis of 34,907 OGLE Systems

Detached eclipsing binary (EB) systems are crucial for measuring the physical properties of stars that evolve independently. Large-scale time-domain surveys have released a substantial number of light curves for detached EBs. Utilizing the Physics of Eclipsing Binaries package in conjunction with Markov Chain Monte Carlo (MCMC) methods for batch parameter derivation poses significant computational challenges, primarily due to the high computational cost and time demands. Therefore, this paper develops an efficient method based on the neural network model and the stochastic variational inference method (denoted NNSVI) for the rapid derivation of parameters for detached EBs. For studies involving more than three systems, the NNSVI method significantly outperforms techniques that combine MCMC methods in terms of parameter inference speed, making it highly suitable for the batch derivation of large numbers of light curves. We efficiently derived parameters for 34,907 detached EBs, selected from the Optical Gravitational Lensing Experiment catalog and located in the Galactic bulge, using the NNSVI method. A catalog detailing the parameters of these systems is provided. Additionally, we compared the parameters of two double-lined detached EBs with those from previous studies and found the estimated parameters to be essentially identical.

An extended variational method for the resistive wall mode in toroidal plasma confinement devices

The external-kink stability of a toroidal plasma surrounded by a rigid resistive wall is investigated. The well-known analysis of Haney and Freidberg is rigorously extended to allow for a wall that is sufficiently thick that the thin-shell approximation does not necessarily hold. A generalized Haney–Freidberg formula for the growth-rate of the resistive wall mode is obtained. Thick-wall effects do not change the marginal stability point of the mode but introduce an interesting asymmetry between growing and decaying modes. Growing modes have growth-rates that exceed those predicted by the original Haney–Freidberg formula. On the other hand, decaying modes have decay-rates that are less than those predicted by the original formula. The well-known Hu–Betti formula for the rotational stabilization of the resistive wall mode is also generalized to take thick-wall effects into account. Increasing wall thickness facilitates the rotational stabilization of the mode, because it decreases the critical toroidal electromagnetic torque that the wall must exert on the plasma. On the other hand, the real frequency of the mode at the marginal stability point increases with increasing wall thickness.

Boundary value problem of calculating ray characteristics of ocean waves reflected from coastline

A direct variational method for solving the problem of reflection of ocean waves ray paths from the coastline with given positions of the source and the point of registration is considered. It is shown that the original boundary value problem can be reduced to the direct search of stationary points of the functional. Information about the objective function in the area of solutions of the ray problem allows us to construct a systematic procedure for finding minima and saddle points. A feature of the proposed approach is the optimization of the ray reflection point along a given coastline.

Detailed derivation of the $\mathbf{^3P_0}$ strong decay model applied to baryons

Abstract We provide an in-detail derivation, through the $^3P_0$ pair creation model, of the transition matrix for a baryon decaying into a meson baryon system. The meson's analysis was successfully conducted in Ref.~\cite{Segovia:2012cd} and we extend the same formalism to the baryon sector, focusing on the $\Delta(1232)\to \pi N$ strong decay width because all hadrons involved in the reaction are very well established, the two hadrons in the final state are stable avoiding further analysis, all quarks are light and so equivalent, and the decay width of the process is relatively well measured. Taking advantage of a very common Rayleigh-Ritz variational method for solving the 2- and 3-body Sch"odinger bound-state equation in which the hadron's radial wave functions are expanded in terms of a Gaussian basis, the expression of the invariant matrix element can be related with the mean-square radii of hadrons involved in the decay. We use their experimental measures in such a way that only the strength of the quark-antiquark pair creation from the vacuum is a free parameter. This is then taken from our previous study of strong decay widths in the meson sector~\cite{Segovia:2012cd}, obtaining a quite compatible result with experiment for the calculated $\Delta(1232)\to \pi N$ decay width and indicating a possible path towards a unified and global description of the baryon and meson strong decay widths within the same constituent quark model framework. Finally, this research work has been developed in setting the stage for a novel raft of applications to exotic hadrons, \emph{i.e.} the description of baryon's coupling to meson-baryon thresholds, one of the mechanism that is thought to be responsible of providing either a large renormalization to naive states or genuine dynamically-generated meson-baryon molecules.Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Article funded by SCOAP3 and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Science and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd